对合交换剩余格类中的合并性
Amalgamation in classes of involutive commutative residuated lattices
AI总结：
该研究通过阿贝尔群簇表示分析对合交换剩余格合并性的三类阻碍，确定固定正幂等元数且层群可除的幂等对称链类满足合并性，同时证明相关半线性簇中合并性普遍失效。
摘要：
我们通过线性有序阿贝尔群簇的范畴表示，研究奇、偶对合交换剩余链中的合并性。该表示分离出三类阻碍因素：离散$κ_J$层引入了带正规嵌入的离散阿贝尔有序群的合并性失效问题；即便$κ_J$为空，特殊层子群仍会在无限制的幂等对称类中阻碍合并性；此外，当两个目标结构以不同方式扩大正幂等骨架时，所有诱导层嵌入的纯度并不足以保证合并性成立。\n随后我们确定了一类充分的正性情形。对任意有限$n≥1$，幂等对称奇链、带幂等假值的幂等对称偶链及其并集，在进一步限制为恰含$n$个正幂等元且典范层群可除的代数后，均满足合并性质（Amalgamation Property，简称AP）。固定的$n$确定了骨架，可除性保证层嵌入为纯嵌入，普通阿贝尔群的推出仍无挠，且同时序扩张定理可构造出一个合并簇。\n尽管强合并性质（Strong Amalgamation Property，简称SAP）的失效可由AP的失效形式化推出，我们仍给出了一个独立的单层例证，证明无限制幂等对称奇、偶类中SAP失效。在簇层面，核心反例给出了半线性簇中AP失效的一般转移准则；具体而言，由幂等对称奇链和偶链生成的簇，以及所有包含奇半线性对合交换剩余格簇的半线性簇，AP均不成立。最后，尽管本文研究的每个固定$n$、层可除的链类都具有AP，但在它们生成的簇中，仅奇单层簇具有AP。
英文摘要：
We study amalgamation in odd and even involutive commutative residuated chains through their categorical representation by bunches of linearly ordered abelian groups. The representation separates three obstructions. A discrete $κ_J$-layer imports the failure of amalgamation for discrete abelian ordered groups with normal embeddings. Even when $κ_J$ is empty, distinguished layer subgroups obstruct amalgamation in the unrestricted idempotent-symmetric classes. Moreover, purity of every induced layer embedding does not suffice when the two targets enlarge the positive-idempotent skeleton in different ways. We then isolate a sufficient positive regime. For every finite $n\geq1$, the idempotent-symmetric odd chains, the idempotent-symmetric even chains with idempotent falsum, and their union have the Amalgamation Property after further restriction to algebras with exactly $n$ positive idempotents and divisible canonical layer groups. Fixed $n$ identifies the skeletons, divisibility makes the layer embeddings pure, ordinary abelian-group pushouts remain torsion-free, and a simultaneous order-extension theorem produces an amalgamating bunch. Although failure of SAP already follows formally from failure of AP, we also give an independent one-layer witness to failure of SAP in the unrestricted idempotent-symmetric odd and even classes. At the variety level, the essential counterexamples yield a general transfer criterion for failure of AP in semilinear varieties; in particular, AP fails for the varieties generated by the idempotent-symmetric odd and even chains, and for every semilinear variety containing the variety of odd semilinear involutive commutative residuated lattices. Finally, although every fixed-$n$, layer-divisible chain class considered here has AP, among the varieties they generate only the odd one-layer variety has AP.
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